Chapter 19: Inference for a Single Mean
Overview
- 19.1 Bootstrap confidence interval for a mean
- 19.1.1 Observed data
- 19.1.2 Variability of the statistic
- 19.1.3 Bootstrap SE confidence interval
19.1.4 Bootstrap percentile confidence interval for a standard
deviation
- 19.1.5 Bootstrapping is not a solution to small sample sizes!
- 19.2 Mathematical model for a mean
- 19.2.1 Mathematical distribution of the sample mean
- 19.2.2 Evaluating the two conditions required for modeling ¯x
- 19.2.3 Introducing the t-distribution
- 19.2.4 One sample t-intervals
- 19.2.5 One sample t-tests
19.1 Bootstrap confidence interval for a mean
- We use bootstrapping to estimate the sampling distribution of a
statistic.
- The process of bootstrapping for a sample mean is the same as
bootstrapping for a sample proportion.
- Since our data is numeric, we calculate the sample
mean \(\bar{x}\), instead of the
sample proportion \(\hat{p}\).
19.1.2 Variability of the statistic
19.1.3 Bootstrap SE confidence interval
Once we have a bootstrapped sampling distribution, we can use it to
make confidence intervals.
- The 95% percentile confidence interval is the the interval
from the 2.5%ile to the 97.5%ile.
- The 95% SE confidence interval is the quick approximation
\(\mbox{point estimate} \pm 2 \cdot
SE_{BS}\)
- We are using 2 instead of 1.96.
19.1.5 Bootstrapping is not a solution to small sample sizes!
- Note: Bootstrapping (and other statistical models) work
best for larger samples.
- The car price example had a sample size of \(n=5\), which is pretty small.
Group Activity
price
18300
20100
9600
10700
27000
Open the Bootstrapping
Applet. Clear the preloaded data and paste in the above
price data (including the header).
Check Show Sampling Options and make a bootstrap
distribution of sample means from this sample.
Use the SD from the distribution to calculate a 95% SE confidence
interval for the mean price.
Select Beyond from the count samples dropdown. Use trial
and error to find the upper endpoint of a 95% percentile confidence
interval. (You want the total area in both tails to be 0.05.)
19.2 Mathematical model for a mean
19.2.1 Mathematical distribution of the sample mean
19.2.2 Evaluating the two conditions required for modeling ¯x
19.2.3 Introducing the t-distribution
19.2.4 One sample t-intervals
19.2.5 One sample t-tests